一个自然数a,若将其数字重新排列可得到一个新的自然数b,如a恰好是b的3倍,称a为一个希望数有赏证明：如a.c都是希望数，则a×c一定是729的倍数

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一个自然数a,若将其数字重新排列可得到一个新的自然数b,如a恰好是b的3倍,称a为一个希望数有赏证明：如a.c都是希望数，则a×c一定是729的倍数
一个自然数a,若将其数字重新排列可得到一个新的自然数b,如a恰好是b的3倍,称a为一个希望数
有赏
证明：如a.c都是希望数，则a×c一定是729的倍数

一个自然数a,若将其数字重新排列可得到一个新的自然数b,如a恰好是b的3倍,称a为一个希望数有赏证明：如a.c都是希望数，则a×c一定是729的倍数
a和b都是希望数,则a=3n,b=3m,a和n的数字和相等,b和m的数字和相等,
a是3的倍数,所以n是3的倍数,a=3n所以a是9的倍数,
由于a是9的倍数,得n也是9的倍数.a=3n所以a是27的倍数,同理,b也是27的倍数
所以ab是27*27=729的倍数
(希望杯的一道题,从"希望数"就看出来了)

惭愧，a,b除以3以后，a和b的数字排列已经没有关系了。所以，以下错误。
∵ a＝3*b
∴ a的数字和是3的倍数
∵ 自然数b是a各位数字的重新排列
∴ b的数字和是3的倍数
我们将等式 a，b 两边同时除以3
如果a, b的数字和仍然是3的倍数，就再两边同时除以3 …… 直到b的数字和不再是3的倍数为止
当b的数字和不再是3的倍数时，a的...

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1035,3105

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多了。以下是部分结果：
(3105,1035) (7425,2475) (30105,10035) (31050,10350) (37125,12375) (42741,14247) (44172,14724) (71253,23751) (72441,24147) (74142,24714) (74250,24750) (74628,24876) (74925,24975) (82755,27585) (85725,28575) (300105,100035) (301050,100350) (307125,102375) (310284,103428) (310500,103500) (321705,107235) (341172,113724) (342711,114237) (370521,123507) (371142,123714) (371250,123750) (371628,123876) (371925,123975) (372411,124137) (384102,128034) (403515,134505) (405135,135045) (411372,137124) (411723,137241) (415368,138456) (415638,138546) (420741,140247) (423711,141237) (427113,142371) (427410,142470) (427491,142497) (428571,142857) (430515,143505) (431379,143793) (431568,143856) (435105,145035) (436158,145386) (441072,147024) (441720,147240) (449172,149724) (451035,150345) (451305,150435) (461538,153846) (463158,154386) (471852,157284) (475281,158427) (517725,172575) (525771,175257) (527175,175725) (568971,189657) (589761,196587) (620379,206793) (620568,206856) (623079,207693) (692037,230679) (692307,230769) (705213,235071) (705321,235107) (711342,237114) (711423,237141) (712503,237501) (712530,237510) (716283,238761) (719253,239751) (720441,240147) (723411,241137) (724113,241371) (724410,241470) (724491,241497) (728244,242748) (741042,247014) (741420,247140) (742284,247428) (742500,247500) (744822,248274) (746280,248760) (746928,248976) (749142,249714) (749250,249750) (749628,249876) (749925,249975) (755271,251757) (771525,257175) (772551,257517) (814752,271584) (822744,274248) (824472,274824) (827550,275850) (827658,275886) (827955,275985) (829467,276489) (841023,280341) (843102,281034) (847422,282474) (852471,284157) (857142,285714) (857250,285750) (857628,285876) (857925,285975) (862758,287586) (862947,287649) (865728,288576) (892755,297585) (895725,298575) (920376,306792) (923076,307692) (943137,314379) (962037,320679) (962307,320769) (3000105,1000035) (3001050,1000350) (3007125,1002375) (3010284,1003428) (3010500,1003500) (3021705,1007235) (3041172,1013724) (3042711,1014237) (3070521,1023507) (3071142,1023714) (3071250,1023750) (3071628,1023876) (3071925,1023975) (3072411,1024137) (3084102,1028034) (3102084,1034028) (3102840,1034280) (3102894,1034298) (3104379,1034793) (3104568,1034856) (3105000,1035000) (3106458,1035486) (3145068,1048356) (3145608,1048536) (3160458,1053486) (3164508,1054836) (3204549,1068183) (3204954,1068318) (3210705,1070235) (3217050,1072350) (3245049,1081683) (3249504,1083168) (3261708,1087236) (3291705,1097235) (3411072,1137024) (3411720,1137240) (3419172,1139724) (3420711,1140237) (3427110,1142370) (3427191,1142397) (3466746,1155582) (3504249,1168083) (3504924,1168308) (3515238,1171746) (3523851,1174617) (3542049,1180683) (3549204,1183068) (3646674,1215558) (3705021,1235007) (3705210,1235070) (3705291,1235097) (3708261,1236087) (3711042,1237014) (3711420,1237140) (3712284,1237428) (3712500,1237500) (3714822,1238274) (3716280,1238760) (3716928,1238976) (3719142,1239714) (3719250,1239750) (3719628,1239876) (3719925,1239975) (3720411,1240137) (3724110,1241370) (3724191,1241397) (3728214,1242738) (3821472,1273824) (3822714,1274238) (3829167,1276389) (3840102,1280034) (3841020,1280340) (3847122,1282374) (3862917,1287639) (3894102,1298034) (3910437,1303479) (3920454,1306818) (3924504,1308168) ......
这是编程做的，我数学不大好。
你找找规律，用数学方法求出通解才好。
对补充的证明：
因为3|a，so 3|b，so 9|a，so 9|b，so 27|a.
用到:a各位数字之和能被3（或9）整除等价于 a能被3（或9）整除

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1035,3105